Preheat the oven to 400°F (205°C). If using a bottom crust, place it in the pie dish and set aside.

Cook the chicken: poach breasts in simmering water for 12–15 minutes until 165°F (74°C), or use roasted/rotisserie chicken. Shred or dice and set aside.

Melt butter in a large skillet over medium heat. Add onion, carrots, and celery; sauté until softened, about 6–8 minutes. Stir in garlic and thyme for 30 seconds.

Sprinkle flour over the vegetables. Cook, stirring constantly, for 2 minutes to cook out the raw flour taste.

Slowly whisk in chicken stock until smooth, then add milk. Bring to a gentle simmer; the sauce will thicken in 2–4 minutes. Season with salt and pepper to taste.

Stir in shredded chicken and peas. Remove from heat.

Pour the filling into the prepared pie shell (or into a casserole if using only a top crust). Cover with the pie top, crimp edges to seal, and cut vents. Brush with egg wash.

Bake for 30–40 minutes until the crust is golden and the filling is bubbling. If the crust browns too quickly, tent loosely with foil for the last 10–15 minutes.

Let rest 10–15 minutes before serving so the filling sets slightly and avoids runny slices.